In this article, we formalize in Mizar  the definition of embedding of lattice and its properties. We formally define an inner product on an embedded module. We also formalize properties of Gram matrix. We formally prove that an inverse of Gram matrix for a rational lattice exists. Lattice of Z-module is necessary for lattice problems, LLL (Lenstra, Lenstra and Lov´asz) base reduction algorithm  and cryptographic systems with lattice .